According to this video of Aggelos Kiayias, IOHK, with an a0 of 0.3, 77% of the rewards are based on the total delegation and 23% are based on the pool’s pledge.
However we have ran some simulations on the formula:
And the result is almost not affected at all. Even with 0 pledge or maximum pledge, the reward difference is insignificant.
The numbers we used are:
R = 15360000, rewards per epoch
a0 = 0.3
σ' = 0.0000425
s' = 3.125E-07
z0 = 0.0209, Saturation ratio
T = 32000000000, total supply
Delegation = 1360000
Pledge = 10000
Am I missing something? Or does the pledge play only a psychological role on the delegators?
For now, there only is a significant difference in rewards if pledge and saturation are both high.
But the next HF probably will have a chance in the formula so the difference in the lower regions might be greater.
Hello @christop
In this formula a0 reduces maximum possible rewards for the pool. At a0 = 0.3 the max rewards at no pledge is 76.92% of available rewards for that pool. The only way to increase this reward to 100% is to have private pool that is fully saturated and where pledge is 100% of total pool.
I can see that you separated the Pledge and Delegation numbers. Please be aware that σ includes Pledge and Delegation, while s is only Pledge. If you do not increase σ when increasing s you will not have a stable point of reference.
Max saturation is about 64 million. Your pledge amount shown is 10,000. If you are just using numbers below 640,000 (which is 1% of total saturation) you will only see less then 1% shift in rewards.
The way the formula is set up is that what a0 takes away in the first part can be returned by increasing the multiplier effect of σ’ in a second part. Up to a point where 1+a0 effect is canceled out by σ’+modifier.
At this point, with such a low a0, you would have to significantly increase the pledge amount to see large difference in rewards. If you want to run a simulation I suggest you use pledge amounts that are a significant % of the pool, such as 10%, 50% and so on. Then you will have a clearer picture of pledge effect on rewards.
a0 is important modifier and if it gets raised by a lot in a future it will significantly affect pool rewards, so it’s a good idea to keep and eye on any changes.
Hope this helps 
